Question: What is the greatest integer value of $x$ for which $5-4x>17$?
First we isolate $x$ by subtracting 5 from both sides.  This gives \[-4x>12.\]Dividing by $-4$ and remembering to reverse the inequality gives us  \[x<-3.\]The greatest integer that solves this inequality is $\boxed{-4}$.

We can check this.  If we substitute $-4$ into the inequality we get  \[5-4(-4)>17\]or  \[5+16>17.\]This is true.  If we substitute $-3$ we get  \[5+12>17,\]which is false.